Constrains on the Electric Charges of the Binary Black Holes with GWTC-1 Events

Eur. Phys. J. C (2021) 81:769 https://doi.org/10.1140/epjc/s10052-021-09555-1

Regular Article - Theoretical Physics

Constrains on the electric charges of the binary black holes with GWTC-1 events

Hai-Tang Wang1,2,a , Peng-Cheng Li3,4, Jin-Liang Jiang1,2, Guan-Wen Yuan1,2,Yi-MingHu5, Yi-Zhong Fan1,2 1 Key Laboratory of Dark Matter and Space Astronomy, Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210023, People’s Republic of China 2 School of Astronomy and Space Science, University of Science and Technology of China, Hefei 230026, Anhui, People’s Republic of China 3 Center for High Energy Physics, Peking University, No. 5 Yiheyuan Rd, Beijing 100871, People’s Republic of China 4 Department of Physics, State Key Laboratory of Nuclear Physics and Technology, Peking University, No. 5 Yiheyuan Rd, Beijing 100871, People’s Republic of China 5 TianQin Research Center for Gravitational Physics and School of Physics and Astronomy, Sun Yat-sen University (Zhuhai Campus), Zhuhai 519082, People’s Republic of China

Received: 20 May 2021 / Accepted: 16 August 2021 © The Author(s) 2021

Abstract Testing black hole’s charged property is a fasci- case, the discharge process is much slower than for ordinary nating topic in modified gravity and black hole astrophysics. plasma so the charged BHs are viable. So it is meaningful to In the first Gravitational-Wave Transient Catalog (GWTC- study the effect on gravitational wave if electric charges are 1), ten binary black hole merger events have been formally presented in these special theories. reported, and these gravitational wave signals have signifi- At present, using the shadow of supermassive BH to esti- cantly enhanced our understanding of the black hole. In this mate its charge has been developed and widely practiced paper, we try to constrain the amount of electric charge with [11], for example Sgr A* and M87* measured by VLBI [12– the parameterized post-Einsteinian framework by treating the 15]. These proposals are either far from being accurate or electric charge as a small perturbation in a Bayesian way. We model-dependent. Therefore, developing new technique to find that the current limits in our work are consistent with model-dependently test the charged property is needed, and the result of Fisher information matrix method in previous gravitational wave has always been expected, which encodes works. We also develop a waveform model considering a the BH’s information [16]. Fortunately, after decades of hard leading order charge effect for binary black hole inspiral. work, LIGO-Virgo announced the first detection of the gravitational wave (GW) signal GW150914 in 2015, generated by the merger of a binary BH [17]. To characterize GW, 1 Introduction we often use inspiral, merger and ringdown to describe the whole coalescence of a binary BH, where the inspiral stage is The prominent black hole (BH) no-hair theorems [1,2]imply generally described by post-Newtonian theory [18], merger vacuum astrophysical black hole can be described by a spe- stage is generally approximated by the numerical simulation cial case of Kerr-Newman metric, which could be character- [19], and ringdown stage is described by the BH perturbation ized by its mass, spin and charge only [3–5]. Although astro- theory [20,21]. Some waveform models, such as IMRPhe- physical BHs are usually considered as electrically neutral nomPv2 (IMR) waveform model, have incorporated these due to charge neutralization by astrophysical plasma, quan- three stages [22]. tum discharge effects, and electron-positron pair production The GW waveform will be affected if the two BHs are elec- [6–8], an accurate upper limit on the amount of the charges of trically charged. For simplicity, consider the electric dipole BHs is still absent. Therefore, it is important and meaningful radiation during their orbital motions, which will be reflected to give a quantitative measurement on the amount of charges in the phase of inspiral stage, and thus provides us a new trick of BHs. Besides, many novel charging mechanisms has been to detect charged binary BH. Depending on the amount of the discussed theoretically, such as primordial black hole [9], charge, there are two different treatments. When the charge and BHs formed by millicharge dark matter [10]. In the later of the black hole is small, the dipole radiation can be taken as a perturbation or correction to the phase of the inspiral a e-mail: [email protected] (corresponding author) stage of the GW waveform, which can be incorporated into

0123456789().: V,-vol 123 769 Page 2 of 7 Eur. Phys. J. C (2021) 81:769 the parameterized post-Einsteinian (ppE) framework [16]. In parameter entering at −1 post-Newtonian order, similar to this case, the correction due to the electric dipole radiation the dipole radiation from the scalar–tensor theory [23]. is completely described by the coefficient of the −1 post- Assuming the early-inspiral stage of the IMR waveform i − Newtonian (PN) order in the waveform, the rest part of the model is reduced to the form of he−ins( f ) = Ae−ins( f )e e ins , waveform is the same as that of the accurate waveform model then the waveform model due to modifications from different describing the coalescence of two neutral black holes, such modified gravity effects can be written as as the well-known phenomenological waveform model [22]. ie−ins+i This is similar to the dipole radiation caused by the scalar he−ins( f ) = Ae−ins( f )e (1) charges carried by black holes in the scalar-tensor theory [23]. However, if the charge of the black hole is large, the In ppE framework [25,26], ppE framework may not be applicable anymore, as the latter b/3 requires the expansion in PN is always linear, which may not ppE = β(πGM f ) , (2) be assured as a prior. We also consider only the leading order gravitational where β is the amplitude coefficient and b is the exponent quadrupole radiation and the electric dipole radiation, based coefficient. Note that the modification enters at (b +5)/2PN 3/5 on which we call such waveform the leading order charged order for a related modified gravity. M = (m1m2) /(m1 + 1/5 (LOC) one. It is apparent that the LOC waveform has a bad m2) is the chirp mass with component masses m1 and m2, accuracy when applied to the analysis of GW data unless the and f is the frequency of the GW. inspiral duration is long enough and the orbit of the two black The study in Tahura et al. [27] shows that at least in theo- holes decays very slow. Despite this shortcoming, the LOC ries where the leading corrections enter at negative PN orders, waveform provides us a toy model such that the effect of the the phase-only analyzes can produce sufficiently accurate charge can be demonstrated explicitly without the assump- constraints. In this analysis, we only consider the correction that the charge of the black hole is small. The detailed tion on the phase and neglect the correction to the ampli- calculation of the LOC waveform is shown in Sect. 5. tude. The frequency at the end of this stage is given by In this work, based on IMR waveform model in LIGO- fc = 0.018/[G(m1 + m2)] [26], above which this model Virgo Algorithm Library [24], we use the Bayesian method is calibrated with numerical-relativity data and can not be to test the dipole radiation of GW signals [25,26] with the applied to ppE formalism. ppE framework. As the main conclusion, we find no visible The effect of the charge on the waveform can be taken as charge taking by astrophysical BH. a perturbation term in the phase [10,28], The rest of this paper is organized as follows. In Sect. 2, we introduce the ppE framework. In Sect. 3, we briefly intro- 5 2/5 2 2 −7/3 q =− η ζ κ (πκGM f ) (3) duce the Bayesian method for the GW data processing and 3584 the results are presented in Sect.4. In order to verify the con- 2 sistency of the results obtained from ppE method, in Sect. 5, where η = m1m2/(m1 + m2) is symmetric mass ratio, ζ we introduce the toy model, the LOC waveform, to study represents the difference between the charges√ of the binary the effect of the amount of the charge on the waveform. In BH and is√ defined by ζ =|λ1 − λ2|/ 1 − λ1λ2, where Sect. 6, we summarize the calculation results and present the λi = qi /( Gmi ) is the charge-to-mass ratio and qi is the conclusion, and discuss the deficiencies of this work as well electric charge of a BH. Here κ = 1 − λ1λ2 ≈ 1 since we as what can be done further. We assume c = 1 throughout consider the effects of λ1 and λ2 as small perturbations. In β =− 5 η2/5ζ 2 =− the paper unless otherwise specified. this case, 3584 and b 7, corresponding to Eq. (2). In TAble I of Yunes et al. [25], obtained constraint |β| 2 Effects on gravitational wave signals 1.6 × 10−4 at −1 PN order for GW150914. The relationship between β and ζ is: β =−5/3584η2/5ζ 2. Thus, one can In this section, we introduce the waveform models adopted derive the corresponding constraint ζ 0.45. There are two in this work. We treat the charge effect as a small pertur- ways to enhance the constraint on the charge. Firstly, one bation, which can be well described by the parameterized may expect the increase of the sensitivities of GW detectors, post-Einsteinian framework. In this case, the phase of the and a more stringent constraint on the charge should appear. inspiral part of the charged binary BHs can be incorporated Secondly, a full gravitational waveform model of charged into the ppE formalism, which describes the gravitational binary BH is excepted to give a more stringent constraint on waveforms of theories alternative to general relativity (GR) the charge. This full GW waveform should apply to the coa- in a model-independent way [16,25]. Formally, the effect lescence of two black holes carried with arbitrary amount of of the electric dipole radiation can be captured by the ppE charge. It is a challenging job and in this work we move for- 123 Eur. Phys. J. C (2021) 81:769 Page 3 of 7 769 ward by developing the leading order charged (LOC) wave- the range of 0.25 ≤ q ≤ 1. We apply comoving uniform prior form model, which is introduced in Sect. 5. on the luminosity distance 50Mpc ≤ dL ≤ 4000Mpc, while the prior on the inclination angle is chosen to be uniform in the range of −1 ≤ cos ι ≤ 1. The prior on the polarization 3 Bayesian inference methods angle is chosen to be uniform in the range of 0 ≤ ψ ≤ π, and the prior of the sky location is chosen to be uniform in To infer the uncertainty of the source parameters θ, which are spherical coordinates. The prior on√ |ζ | is chosen to be uni- quantified by the posterior probability distribution p(θ|d, M), form in the range of 0 ≤|ζ | < 2. For other parameters we perform Bayesian analysis with the prior p(θ|M) and the in the e-insp-ppE waveform model, we use the same priors likelihood with Gaussian noise assumption for the GW data, presented in Abbott et al. [32].

N 1 4 Results of Bayesian inferences p(d|θ, M) ∝ exp − di − hi |di − hi , (4) 2 = i 1 In this section, on the promise that the BH has a small amount of charge, we set up constraints on the dipole radiation with where di is the data of the ith instrument, M is the model assumption, N is the number of detectors in the network of the ppE framework. We analyze with e-insp-ppE waveform Advanced LIGO and Advanced Virgo, and h is the wave- model to constrain the dipole radiation, with a high frequency i = . /[ ( + )] form model calculated with θ for the ith detector. The noise cut-off at fc 0 018 G m1 m2 . Specifically, this weighted inner product a( f )|b( f ) is defined by analysis is applied to GW150914, GW151226, GW170104, GW170608, and GW170814. We neglect other events since fhigh a( f )b( f ) the early-inspiral signal-to-noise ratios (SNRs) of them are a( f )|b( f )= R f, 4 ( ) d (5) all less than 6. f w Sn f lo AsshowninTable1, for GW170608, the GW data can |ζ | . where flow and f figh are the high and low pass cut-off fre- constrain to be less than about 0 21 at 90% credible level, |ζ |≤ . quencies respectively, Sn( f ) is the power spectral density of while the loosest case is given by GW170104, 0 70 the detector noise. The Bayesian theorem is described by (at 90% credible level) due to its lowest SNR of the inspiral signal among all these five events. In Yunes et al. [25], the p(d|θ, M)p(θ|M) authors got constraints on scalar dipole radiation through p(θ|d, M) = , (6) p(d|M) a Fisher parameter estimation study, using a fitted spectral noise sensitivity curve. As a result, the constraint on where p(d|M) = p(d|θ, M)p(θ|M)dθ is the evidence of GW150914 is |ζ | 0.45, and the constraint on GW151226 a specific model assumption. is |ζ | 0.24. Their results agree well with ours shown in There are fifteen parameters in the IMR waveform model, Table 1. One can also get constraints on |ζ | by reweighting including the redshifted chirp mass (Mz), mass ratio (q), the posteriors of parameters δφ−2 from results in Abbott et 2 luminosity distance (dL ), inclination angle (θ jn), the refer- al. [26], where δφ−2 =−5/84ζ is the parameterized vio- ence orbital phase (φc), the geocentric time (tc), the polar- lation of GR at −1 PN. Recently, a similar analysis was per- ization angle (ψ), the two dimensional sky location, and six formed in Wang et al. [33] with both reweighting method and spin parameters. And the early-inspiral stage of the IMR Bayesian inference method, and they find that the Bayesian waveform model for ppE framework (hereafter e-insp-ppE analysis is more reliable than the reweighting analysis. waveform model) has an additional ppE parameter ζ .We To further discuss the reliability of our results, we apply marginalize over the reference phase φc and the geocentric Bayesian inference on GW data of GW150914 with a set of time tc, thus we have 14 free parameters for e-insp-ppE wave- low-pass cut-off frequencies fhigh. And we show the chirp form model. The GW data and power spectral density for each mass distribution in Fig. 1, in which the distribution becomes event are downloaded from LIGO-Virgo GW Open Science wider as fhigh less than 400 Hz. Nevertheless, the result is Center [29]. To estimate parameters with data from the first not biased even the cut-off frequency is as low as 50 Hz, two observation runs (O1 and O2), we carry out Bayesian although the uncertainty is a bit larger. inference with Bilby [30], using Pymultinest [31]asour Actually, the charge will significantly affect the space- sampler. time metric of the binary system if the charge is as large The prior on the redshifted chirp mass is chosen to be uni- as the maximum allowable value. This effect will be fully form in the range of 5 M ≤ Mz ≤ 20 M for GW151226 reflected in waveform model when higher PN order terms and GW170608 while 5 M ≤ Mz ≤ 50 M for the other are taken into account. However, an analytic solution will not events, the prior on the mass ratio is chosen to be uniform in be available in the forthcoming future and one can only get 123 769 Page 4 of 7 Eur. Phys. J. C (2021) 81:769

Table 1 The constraints on Event name GW150914 GW151226 GW170104 GW170608 GW170814 dipole radiation of ﬁve selected binary BH events, 90% fc[Hz] 50 153 60 179 58 conﬁdence intervals are shown |ζ | . +0.22 . +0.10 . +0.32 . +0.07 . +0.15 0 20−0.17 0 16−0.10 0 38−0.29 0 14−0.07 0 12−0.10

form model, we do not consider the spin of binary black holes since the spin evolution of the binary BH only occurs at high orders of the PN expansion [40,41]. Therefore, we only need to analyze the orbit (circular) evolution in the inspiral phase due to the energy loss. For two point particles with mass m1 and m√2 and charge q1 and q2 respectively, we deﬁne λi = qi /( Gmi ). They orbit each other with orbital radius R, and the orbit decays with time t. The dissipation of total energy can be divided into two parts, one is the emission of GW, the other is electromagnetic dipole radiation, which can be written as [10] dE dE =−dEGW − dip dt dt dt GG3 M5 G3 m2m2 =−32η2 eff − 2ζ 2 eff 1 2 , 5 4 (7) Fig. 1 The marginalized posterior probability distributions of chirp- 5 R 3 R = mass of GW150914, with different high frequency cut-off fhigh = + 50, 100, 200, 400, 600, 800, and 1024 Hz. The uncertainty of the dis- where M m1 m2 is total mass of the binary, = ( − λ λ ) ζ = tribution becomes larger as fhigh is less than 400 Hz Geff G √1 1 2 is the effective Newton constant, |λ1 − λ2|/ 1 − λ1λ2 is used to characterize the difference 2 between the charges of two BHs, and η = m1m2/(m1 +m2) it numerically, i.e. Bozzola and Paschalidis [34]. In Bozzola is symmetric mass ratio. The evolution equation of the orbital and Paschalidis [34], the binary charged black holes were radius arising from Eq. (7)is studied via numerical general relativity method, where the higher PN order effects of the charge are included and the dR(t) A B − = + , (8) amount of charge is set free. One of the main conclusions dt R(t)3 R(t)2 from this paper is that the greatest difference between charged and uncharged black holes arises in the earlier inspiral. This = 2 / = ζ 2 2 / where A 64GGeff Mm1m2 5 and B 4 Geff m1m2 3. provides a convincing evidence to support the reasonability When B is not equal to zero, the relation between R and t of our work. can be parameterized as For future space-based GW detectors such as Laser Inter- ferometer Space Antenna (LISA) [35] and TianQin [36], the 1 A A2 A3 B duration of GW strain from BH binaries are much longer τ = R3 − R2 + R − ln R + 1 3B 2B B3 B4 A [37,38]. Since the dipole gravitational radiation dominates at −1 PN order, it is more suitable for detecting by LISA and R4 1 B B2 B3 B4 − R + R2 − R3 + R4 TianQin. Barausse et al. [39] shows that joint observations A 4 5A 6A2 7A3 8A4 of GW150914-like systems by LIGO-Virgo and LISA will BR 9 + O (9) improve bounds on dipole emission from BH binaries by sev- A eral orders of magnitude relative to current constraints. We expect that with multi-band GW the constraint on the electric where τ = t − t and t is the coalescence time. It is clear charge of BH can be improved as well. c c that an analytical solution of R(t) is not possible in Eq. (9), so we have to solve it numerically. Due to the last part of Eq. (9), the numerical solution is not accurate enough when 5 The leading order charged waveform BR/A ∼ 0, thus we expand Eq. (9)asaseriesofBR/A . According to the Kepler’s law, the orbital frequency is 3 In this section, we try to learn more about the effect of the Geff M/R and the gravitational frequency is twice as charge by developing a LOC waveform model. For this wave- much as the orbital frequency, so we have ωgw = 2π fgw = 123 Eur. Phys. J. C (2021) 81:769 Page 5 of 7 769

3 2 Geff M/R . The waveform in time domain contains two of the GW signal will be affected significantly if GW150914 parts [41,42] is endowed with a large amount of charges. The effect of 5/3 2/3 2 charge on the early inspiral of the gravitational waveform 4 Geff M π fgw (t) 1 + cos ι h+(t) = ( (t)) , 2 cos is more significant, which is consistent with previous works dL c c 2 M 5/3 2/3 [10,16,25]. It is also worth noting that Geff and have sim- 4 Geff M π fgw (t) h×(t) = cos ι sin ( (t)) , ilar effects on shaping the GW signals, the smaller Geff ,the d c2 c L lower GW amplitude, as described in Eq. (10). We also find (10) that the larger value of λ1λ2, the slower frequency evolution where ι is inclination angle, dL is luminosity distance and of GW signals. the phase of waveform is t 6 Summary (t + tISCO) = dt ωgw t + φc, (11) tISCO In this work, we study the charge effect on the inspiral stage where tISCO is the time when the BH reaches the innermost of the GW waveform. By considering the charge effect as an stable circular orbit (ISCO). We cut the phase before tISCO perturbation in the inspiral stage, we perform Bayesian infer- for three reasons: first, the phase before tISCO can be included ence on five events of the first Gravitational-Wave Transient in φc and the effect is equal to a time shift, which means this Catalog (GWTC-1), GW150914, GW151226, GW170104, does not affect the results of parameter estimation; second, GW170608, and GW170814, whose SNRs are larger than ωgw is infinity when τ = 0, and this can not be integrated; last 6 after applying a low-frequency cut off fc. Based on the but not least, the LOC waveform model cannot truly describe ppE formalism, the most stringent limit from FIM up to now the motion of the binary BH when the orbital distance is too is |ζ |∼0.24 [10,25], while our Bayesian based result is small. |ζ |∼0.21 at the 90% credible level, which is given by According to Eqs. (8) and (9), tISCO is fixed for a given GW170608. The analysis on GW151226 and GW170814 RISCO. In reality, it is difficult to know the exact values of M give similar constraints, i.e., ζ<0.26 and ζ<0.27 at 90% and λ for the final BH. Because with a given λ1λ2 and |ζ |,we credible level respectively. For GW150914, the constraint still cannot uniquely determine the respective charge of the is ζ<0.42 at 90% credible level, which is similar to the two BHs. For example if λ1 = a, λ2 = b is the solution then analysis before [25]. λ1 = b, λ2 = a could still be the solution. Approximately To explicitly show the effect of the charge on the wave- λ + λ λ + λ we take M = m +m ,λ= min{| m1 1 m2 2 |, | m1 2 m2 1 |}. form, we developed the LOC waveform model in Sect. 5. 1 2 m1+m2 m1+m2 then in the similar way to Pugliese et al. [43] we obtain Although due to the lack of accuracy, this waveform cannot be used to analyze the realistic GW data, this toy model has 4GMλ2 the advantage of being adjustable and intuitive. The LOC R = , (12) ISCO 3 + 1/C + C waveform is obtained by taking into account the dissipa- √ tion effect caused by the electric dipole radiation and the where C =−(9 − 8λ2 − 4 4λ4 − 9λ2 + 5)1/3.TheISCO quadrupole gravitational radiation on the circular orbit of of a charged BH decreases with the charge, and we have the charged binary BHs. Like the 0PN waveform model, the RISCO = 4GM for |λ|=1 and RISCO = 6GM for the spin of BH is not considered for the LOC waveform model non-rotating uncharged BH, respectively. as which emerges in the waveform only at higher orders. Based on the discussion above, we obtain the LOC wave- Besides, the analytical LOC waveform is achievable only if form model. Obviously, it reduces to the 0PN when both the charge is treated as a correction, so the general charge BHs are uncharged. As we have already pointed out, under case is gotten numerically. We find that both the amplitude the ppE framework, LIGO-Virgo has very weak restrictions and phase of the GW signal will be strongly affected if the on the charges of GW150914, with |ζ | 0.45. Here we BHs are endowed with a large amount of charge. show that such large amount of charge could have non- The work in this paper can be improved in several aspects. negligible effect on the waveform. We choose parameters For example, in the employment of the ppE framework, we similar to GW150914, where m1 = m2 = 30 M, dL = only consider the electric dipole radiation, so when the two 540Mpc,ι= 0,φc = 0, and we set low cut-off frequency to black holes have the same charge-to-mass ratio, the charge be flow = 20 Hz. For the charge of GW150914, if |ζ |=0.45, effect disappears in the phase of the ppE waveform. To over- then possible allowed range of λ1λ2 is (−0.053, 0.80).For come this problem, one can consider the electric quadrupole proper comparison, we have chosen four special values with radiation. Moreover, as we have shown above, the constraints λ1λ2 ={−0.05, 0.05, 0.15, 0.25} to study the effect of on the electric charges of the black holes are still not very charge. As shown in Fig. 2, both the amplitude and phase stringent, one of the possible ways to improve this is to add 123 769 Page 6 of 7 Eur. Phys. J. C (2021) 81:769

Fig. 2 The tapered LOC gravitational waveforms for BH binaries with of zero charge, and the red (blue) dash dotted (dashed) line in the left different charges. The rest parameters of these waveforms are the same, panel represents the waveform with parameter λ1λ2 =−0.05 (0.05), i.e., the detector-frame masses are m1 = m2 = 30 M, the luminosity the red (blue) dash dotted (dashed) line in the right panel represents the distance is dL = 540 Mpc, the inclination angle is θ jn = 0. The low cut- waveform with parameter λ1λ2 = 0.15 (0.25) off frequency is flow = 20 Hz. The green solid line represents the case

higher order corrections to the waveform. However, this may article are analyzed by using analytical formulas combined with public not be helpful, because the parameterized constraints on the software, so no data is provided to readers. If someone needs relevant PN coefﬁcients obtained by LIGO-Virgo show that the -1 posterior distribution, they can contact the corresponding author.] PN gets the most stringent constraint than other higher PN Open Access This article is licensed under a Creative Commons Attri- orders [26], which we expect also applies to the study of bution 4.0 International License, which permits use, sharing, adaptation, the charged black holes. The contributions from the higher distribution and reproduction in any medium or format, as long as you orders corrections of some other speciﬁc theories has been give appropriate credit to the original author(s) and the source, pro- vide a link to the Creative Commons licence, and indicate if changes studied in Yunes et al. [25]. were made. The images or other third party material in this article Above all, we give the constraints on the dipole radia- are included in the article’s Creative Commons licence, unless indi- tions of GW events from GWTC-1. These constraints can cated otherwise in a credit line to the material. If material is not be also converted to the constraints on radiation effects in included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permit- other forms, such as the radiation caused by the magnetic or ted use, you will need to obtain permission directly from the copy- other U(1) dark charges carried by black holes. In the future, right holder. To view a copy of this licence, visit http://creativecomm similar analysis can be applied to GW data that published in ons.org/licenses/by/4.0/. 3 GWTC-2 [44]. Funded by SCOAP .

Acknowledgements We sincerely thank Vitor Cardoso and Laura Bernard for their kindhearted help. Hai-Tian Wang also appreciates Yi-Fu Cai, Jian-dong Zhang, Bing-bing Zhang, Si-ming Liu, Can- References ming Deng, Ming-Zhe Han, and Shao-Peng Tang for insightful com- ments and discussions. This work has been supported by NSFC under 1. S.W. Hawking, Commun. Math. Phys. 25, 152 (1972) grants of No. 11525313 (i.e., Funds for Distinguished Young Schol- 2. D.C. Robinson, Phys. Rev. Lett. 34, 905 (1975) ars), No. 11921003, No. 11335012, No. 11325522, No. 11735001, 3. W. Israel, Commun. Math. Phys. 8, 245 (1968) No. 11703098, No. 11847241 and No. 11947210. Peng-Cheng Li 4. B. Carter, Phys. Rev. Lett. 26, 331 (1971) is also funded by China Postdoctoral Science Foundation Grant No. 5. V. Cardoso, L. Gualtieri, Class. Quantum Gravity 33, 174001 2020M670010. This research has made use of data, software and/or (2016). arXiv:1607.03133 [gr-qc] web tools obtained from the Gravitational Wave Open Science Cen- 6. P. Goldreich, W.H. Julian, Astrophys. J. 157, 869 (1969) ter (https://www.gw-openscience.org), a service of LIGO Laboratory, 7. G.W. Gibbons, Commun. Math. Phys. 44, 245 (1975) the LIGO Scientiﬁc Collaboration and the Virgo Collaboration. LIGO 8. M.A. Ruderman, P.G. Sutherland, Astrophys. J. 196, 51 (1975) is funded by the U.S. National Science Foundation. Virgo is funded 9. L. Liu, Z.-K. Guo, R.-G. Cai, S.P. Kim, Phys. Rev. D 102, 043508 by the French Centre National de Recherche Scientiﬁque (CNRS), the (2020). arXiv:2001.02984 [astro-ph.CO] Italian Istituto Nazionale della Fisica Nucleare (INFN) and the Dutch 10. V. Cardoso, C.F.B. Macedo, P. Pani, V. Ferrari, J. Cosmol. Nikhef, with contributions by Polish and Hungarian institutes. Astropart. Phys. 2016, 054 (2016). arXiv:1604.07845 [hep-ph] 11. T. Johannsen, A.E. Broderick, P.M. Plewa, S. Chatzopoulos, S.S. Data Availability Statement This manuscript has no associated data Doeleman, F. Eisenhauer, V.L. Fish, R. Genzel, O. Gerhard, M.D. or the data will not be deposited. [Authors’ comment: The results in the Johnson, Phys. Rev. Lett. 116, 031101 (2016) 123 Eur. Phys. J. C (2021) 81:769 Page 7 of 7 769

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